Most essays written in an academic setting fall into one of four categories, or modes: exposition, narration, description, and persuasion. There are variations (or subcategories) of different essays that are written in each main mode, but each variation ultimately has the same overall purpose:
Exposition: an essay that attempts to inform the reader about something important or explain something to him/her (a process, a set of rules, the benefits of an activity, etc.). Common expository writings: business or technical writing, process writing, compare and/or contrast essays, reaction essays, response essays, and often research-based essays.
Narration: writing that tells the reader about a particular event(s) that took place. Common narration writings: personal essays, short stories, novels, poetry.
Description: writing that uses vivid language to describe a person, place, or event so that the reader can picture the topic clearly in his/her mind. Fiction and poetry often use large amounts of descriptive writing, and sometimes only attempt to serve this descriptive purpose.
Persuasion: writing that takes a stand on a principle (oftentimes a controversial issue) and attempts to persuade the reader to adopt a similar mindset. At the collegiate level, persuasive writing assignments are quite common. The most common persuasive essay is the argument paper. The reason for this is because writing an argument essay involves incorporating critical thinking and often the use of outside sources. Many papers written in disciplines other than English are a variation of the argument paper and should be treated in a similar fashion (ex-an analysis essay for literature that defends a particular form of literary criticism for the literary work).
It is important to note that while an essay generally falls under one of these main modes, good writing usually incorporates a variety of these into the same assignment. For instance, a narration piece will benefit from incorporating heavy amounts of description, and often a persuasive essay must first explain certain facts to the reader (exposition) before arguing for or against a certain solution.
Writers of all disciplines and backgrounds should study these modes and learn how to write each one successfully and incorporate them into different forms of writing.